ICR QUARTERLY REPORT NO. 27. Page: 42 of 127
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If A-Z9
Proof: Let C kid Then we have C R,, with RC IA],
universal from C to the inclusion. Since A C B, of course RC e B ,
and we have to show C? is still universal from C to the inclusion
of B in C. So let g:C-'--B, B CI]B], and let A be an object
of A isomorphic to B, j:BS A, using equivalence. Then there is
a unique f:RC- A such that f ' = jg. We conclude that there is
a unique U':RC P B in B such that it N C = g, namely f' = ,
using fullness. For if also f":RC--B satisfies f" r = g, then
(jfr) = jg, so that jf' = jf', or f" = since j is an isomorphism.
C RC
g
ig
A
A much simpler but more sophisticated proof uses the
composition of adjoint situations (see (8. ]).
The following fact is useful in considering various restrictions
of the main adjunctions in the body of this paper.
Lemma2: If F:A ---- B, G:1B- A, F -j , and if A and B
=0 =
are full subcategories of A and B (respectively) such that the
restriction of F to A factors through B and the restriction of
G to B 'factors through , yielding F A -9B and G :B -'A
(respectively), then F0 H G.
Proof: The restriction of the natural isomorphism on A" B
to A)( B yields the desired natural isomorphism , noting
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Ashenhurst, R.L. ICR QUARTERLY REPORT NO. 27., report, January 1, 1970; United States. (https://digital.library.unt.edu/ark:/67531/metadc871539/m1/42/: accessed May 3, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.